Reasoning under Uncertainty: Intro to Probability

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Reasoning under Uncertainty Intro to Probability
Computer Science cpsc322, Lecture 24 (Textbook
Chpt 10.1) March, 10, 2008
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Announcements

• Assignment 3
• has been out for a few days. It is due on the
  19th. Make sure you start working on it soon.
• One question requires you to use datalog (with TD
  proof) in the AIspace.
• To become familiar with this applet download and
  play with the simple examples we saw in class.
• Slides with notes remember that I always post my
  slides with annotation after class
• Midterm maximize your learning

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Lecture Overview

• Big Transition
• Intro to Probability
• .

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Big Transition
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Reasoning under Uncertainty Diagnosis
Bayes Net to diagnose liver diseases
Source Onisko et al., 99
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Reasoning Under Uncertainty
Decision Network in Finance for venture capital
decision
Source R.E. Neapolitan, 2007
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Modules we'll cover in this course

• Environment

Stochastic
Deterministic
Search
Single-stage Decision Networks
Single Action
Constraint Satisfaction (CSPs)
Bayes Nets
Decision
Logics
Multi-stage Decision Networks
Search
Sequence of Actions
Constraint Satisfaction (CSPs)
Markov Decision Processes (MDPs)
Planning
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Planning Under Uncertainty
Learning and Using POMDP models of
Patient-Caregiver Interactions During Activities
of Daily Living
Goal Help Older adults living with cognitive
disabilities (such as Alzheimer's) when they

• forget the proper sequence of tasks that need to
  be completed
• they lose track of the steps that they have
  already completed.

Source Jesse Hoey UofT 2007
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Planning Under Uncertainty

• Helicopter control MDP, reinforcement learning
• States all possible positions, orientations,
  velocities and angular velocities
• Final solution involves
• Deterministic search!

Source Andrew Ng 2004
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Lecture Overview

• Big Transition
• Intro to Probability

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Intro to Probability (Motivation)

• AI agents (and humans ?) are not omniscient.
• The problem is not only predicting the future or
  remembering the past
• Is it raining outside now?
• How many people are in this building?

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Intro to Probability (Key points)

• Outside it is raining or it is not
• Agents are not all ignorant to the same degree
• Agents have to act in spite of ignorance (not
  acting usually has implications)
• So agents need to represent their ignorance

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Probability as a formal measure of
uncertainty/ignorance

• Belief in a proposition f can be measured in
  terms of a number between 0 and 1 this is the
  probability of f
• The probability f is 0 means that f is believed
  to be definitely false.
• The probability f is 1 means that f is believed
  to be definitely true.
• Using 0 and 1 is purely a convention.
• Note you can be certain, but you can be wrong!

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Random Variables

• A random variable is a variable that is randomly
  assigned one of a number of possible values
  (exhaustive and mutually exclusive)
• The domain of a random variable X, written
  dom(X), is the set of values X can take
• Examples (boolean and discrete)

• A tuple of random variables ltX1 ,., Xngt is a
  complex random variable with domain..

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Random Variables (cont)

• Assignment Xx means X has value x
• A proposition is a Boolean formula made from
  assignments of values to variables
• Examples

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Possible Worlds

• A possible world specifies an assignment to each
  random variable

• E.g., if we model only two Boolean variables
  Cavity and Toothache, then there are 4 distinct
  possible worlds
• Cavity T ?Toothache T
• Cavity F ? Toothache T
• Cavity T ? Toothache F
• Cavity T ? Toothache T

• Notice Possible worlds are mutually exclusive
  and exhaustive

w Xx means variable X is assigned value x in
world w
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Semantics of Probability

• The belief of being in each possible world w can
  be expressed as a nonnegative measure µ(w)

• For sure, I must be in one of themso

SW µ(w) 1

• What is the probability of a proposition f ?
• It can be computed as the sum of µ(w) for all
  the worlds in which f is true
• P(f)S w f µ(w)

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Semantic of probability Example

• µ(w) for possible worlds generated by three
  Boolean variables toothache, catch, cavity
• For any f, sum the prob. of the worlds where it
  is true
• P(f)S w f µ(w)

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Semantic of probability Complex Propositions

• µ(w)
• For any f, sum the prob. of the worlds where it
  is true
• P(f)S w f µ(w)
• Ex P(toothache)

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Semantic of probability Complex Propositions

• µ(w)
• For any f, sum the prob. of the worlds where it
  is true
• P(f)S w f µ(w)
• P(cavity or toothache) 0.108 0.012 0.016
  0.064
• 
  0.0720.08 0.28

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Probability Distributions

• A probability distribution P on a random variable
  X is a function dom(X) - gt 0,1 such that
• x -gt P(Xx)

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Joint Probability Distributions

• When we have multiple random variables, their
  joint distribution is a probability distribution
  over the variable Cartesian product
• E.g., P(ltX1 ,., Xngt )
• Think of a joint distribution over n variables as
  an n-dimensional table
• Each entry, indexed by X1 x1,., Xn xn
  corresponds to P(X1 x1 ? . ? Xn xn )
• The sum of entries across the whole table is 1

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Summary

• Probability (informal)
• Random variables
• Possible Worlds
• Probability (formal)
• Probability Distributions

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Next Class

• Marginalization
• Conditional Probability
• Chain Rule
• Bayes' Rule
• Independence